We present a photometric investigation of the variation in galaxy colour with environment in 11 X-ray-luminous clusters at 0.07 ≤ z ≤ 0.16 taken from the Las Campanas/AAT Rich Cluster Survey. We study the properties of the galaxy populations in individual clusters, and take advantage of the homogeneity of the sample to combine the clusters together to investigate weaker trends in the composite sample. We find that modal colours of galaxies lying on the colour-magnitude relation in the clusters become bluer by d(B - R)/drp = -0.022 ± 0.004 from the cluster core out to a projected radius of rp = 6 Mpc, further out in radius than any previous study. We also examine the variation in modal galaxy colour with local galaxy density, Σ, for galaxies lying close to the colour-magnitude relation, and find that the median colour shifts bluewards by d(B - R)/dlog10(Σ) = -0.076 ± 0.009 with decreasing local density across three orders of magnitude. We show that the position of the red envelope of galaxies in the colour-magnitude relation does not vary as a function of projected radius or density within the clusters, suggesting that the change in the modal colour results from an increasing fraction of bluer galaxies within the colour-magnitude relation, rather than a change in the colours of the whole population. We show that this shift in the colour-magnitude relations with projected radius and local density is greater than that expected from the changing morphological mix based on the local morphology-density relation We therefore conclude that we are seeing a real change in the properties of galaxies on the colour-magnitude relation in the outskirts of clusters. The simplest interpretation of this result (and similar constraints in local clusters) is that an increasing fraction of galaxies in the lower density regions at large radii within clusters exhibit signatures of star formation in the recent past, signatures which are not seen in the evolved galaxies in the highest density regions.